On the computation of geometric features of spectra of linear operators on Hilbert spaces (Q6566148)

In this article, the authors study the problem of understanding geometric features of spectra. More precisely, they ask the following interesting question. Suppose that one is given a bounded operator \(A\) in \(B(l^2(\mathbb N))\). Do there exist algorithms that approximate geometric features such as for example, spectral gaps, fractal dimensions and notions of sizes and capacity measures of the set \(\mathrm{S}_p(A)\) from a matrix representation of the operator \(A\)? Indeed, computing spectra is an important problem in computational mathematics with many applications and in these applications it is important to determine geometric features of spectra such as the above. The authors provide algorithms for the computation of many geometric features such as the above.\N\NThese results show that infinite-dimensional spectral problems yield an intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithm. This is very much related to S. Smale's comprehensive program on the foundations of computational mathematics initiated in the 1980s. The authors classify the computation of geometric features of spectra in the SCI hierarchy, allowing them to precisely determine the boundaries of what computers can achieve (in any model of computation) and prove that their algorithms are optimal. The authors also provide a new universal technique for establishing lower bounds in the SCI hierarchy, which both greatly simplifies previous SCI arguments and allows new, formerly unattainable, classifications.\N\NThe paper is very well written and has an excellent set of references.